Besov spaces and Schatten class Hankel operators for Hardy and Paley--Wiener spaces in higher dimensions

TL;DR

This paper characterizes Schatten class membership of Hankel operators on higher-dimensional Paley--Wiener and Hardy spaces using Besov space symbols, extending classical results to more general domains and dimensions.

Contribution

It develops a Besov space framework for Paley--Wiener spaces in higher dimensions and links Hankel Schatten class membership to Besov space membership of symbols.

Findings

Hankel operators belong to Schatten class $S^p$ iff their symbols are in corresponding Besov spaces.

Extension of Schatten class criteria to all $1 \\leq p < \\infty$ for classical Hardy spaces.

Extension of results to Paley--Wiener spaces on bounded smooth domains with positive curvature.

Abstract

We consider Schatten class membership of Hankel operators on Paley--Wiener spaces of convex $\Omega \subset \mathbb{R}^n$, both for bounded and unbounded domains. In particular, the classical product Hardy spaces fit within our theory. For admissible domains, we develop a framework and theory of Besov spaces of Paley--Wiener type, and prove that a Hankel operator belongs to the Schatten class $S^p$ if and only if its symbol belongs to a corresponding Besov space, for $1 \leq p \leq 2$. We extend this result to all $1 \leq p < \infty$ for the classical product Hardy spaces and to $1 \leq p < 2(n+1)/(n-1)$ for the Paley--Wiener space of a bounded smooth domain $\Omega \subset \mathbb{R}^n$ of strictly positive curvature.